JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Proximity Factor in the Judd Color Difference Formula*ALAN C. TRAUBit AND ISAY BALINKINUniversity of Cincinnati, Cincinnati, Ohio

(Received November 12, 1959)

When two colored samples are to be compared visually, thesize of the perceived lightness difference will in general dependupon the spatial proximity of the samples. This is taken intoaccount in the Judd formula for color difference [D. B. Judd,Am. J. Psychol. 52, 418 (1939).] which includes a proximityfactor. This factor weights measured reflectance differences sothat the perceived lightness differences predicted by the formulavary with the spatial proximity of the samples. A few tentativevalues have been assigned by Judd for various separations. Acontinuous, empirical expression for the proximity factor has beenfound as a function of angular separation of the samples underspecified viewing conditions. A divided visual field was used, with

THE PROBLEM

A NY attempt to introduce order into the apparently1 chaotic realm of color sensations involves theselection of a mode of representation and the formula-tion of rules whereby each color sensation is uniquelyassociated with some element of the representation.Because the entire range of visible surface colors maybe described by manipulation of three variables alone,such as hue, saturation, and lightness, an obvious modeof representation is a graphical one in physical space.Each of the three independent color variables may beassociated with a dimension in space, and the positionof each color may be uniquely assigned after a systemof scale units has been designated.

It was such reasoning which led to the idea of thecolor solid, a concept which has proved useful since thetime of Helmholtz.

The proper choice of scale units, and thus the entiregeometry of the color space, has long been a matter ofspeculation. There is nothing in the spectral-reflectancecharacteristics of colored surfaces to suggest the geo-metrical properties which a color space should have. Interms of the human observer, however, an extremelyuseful color space would be one in which the scale unitshave equal visual significance. Equal physical distanceswithin the color space would then signify colorimetri-cally equal distances. Such a color space would bevisually homogeneous and isotropic and would fulfillthe need for a system of color-tolerance specification.

The geometry of a continuum is specified by itsmetric, which in this case becomes a formula for colordifference expressed in terms of the color-space co-ordinates. A typical color-difference formula includesterms for hue difference, saturation difference, and

* Presented at the Spring Meeting of the Optical Society ofAmerica, 1954.

t This is an abstract of the dissertation submitted to theGraduate School of Arts and Sciences of the University ofCincinnati in partial fulfillment of the requirements for the degreeof Ph.D., 1952.L Now at Fenwal Incorporated, Ashland, Massachusetts.

variably separable halves. The halves could be interchangedwithout the knowledge of the observer, who was required toidentify the darker half many times at each of several values ofseparation. The observer's difficulty of identification increasedwith increasing dividing linewidth and was measured statisticallyin terms of errors of identification. From the error data and certainassumptions, a relationship between proximity factor and angularseparation was deduced. A few tests were made in order to studypossible effects of dividing linewidth on chromaticity discrimina-tion. Under the particular test conditions which we used, no effectwas observed.

lightness difference. In many cases, and in this paper,hue and saturation differences are treated together asa single quantity, chromaticness difference. 1 Any colordifference, then, may be treated as the combination ofa lightness difference and a chromaticness difference,and the color solid is conveniently pictured as a stackof horizontal chromaticness planes of increasing light-ness in the upward direction.

Existing color spaces, and there are many, approachvisual uniformity but do not achieve it perfectly.2 Animportant problem is the choice of relative sizes of scaleunits along the different axes, particularly the visualrelationship between a unit of chromaticness differenceand a unit of lightness difference. These differenceshave psychophysical meaning in that they refer tosensation differences corresponding to physically meas-urable stimulus differences. A lightness difference, forexample, is the sensation experienced when one observesa luminous-reflectance difference. The problem ofchoosing these units so that they are visually equal iscomplicated by the fact that the perceptual size of acolor difference depends upon observer and viewingconditions as well as upon the spectral natures of thesamples.

Let us examine in detail the psychophysical nature ofthe lightness axis. What, we ask, shall its metric prop-erties be? An obvious starting point is the psycho-physical quantity luminous reflectance, a property ofall material surfaces. This is the spectrally integratedreflectance of a surface evaluated as a human observerwould evaluate it. It will hereafter be designated simplyas reflectance.

Designating a direction in space as a reflectance axis,let us say the vertical direction, one may choose asegment of the axis and divide it into a number of inter-vals of equal physical length. Let there be, for example,100 intervals with each designated as a unit, such as a

1 The psychological effect of a hue-saturation combination iscustomarily designated as "chromaticness" to distinguish it fromits psychophysical counterpart "chromaticity."

2 R. W. Burnham, J. Opt. Soc. Am. 39, 387 (1949).

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ALAN C. TRAUB AND ISAY BALINKIN

percent. Each point on the axis may be associated witha given value of reflectance, provided that the reflec-tance measuring procedure is likewise defined in termsof one hundred equal intervals. As the next step, the re-flectance scale may now be divided into an arbitrarynumber of intervals of equal visual significance. It is herethat the distinction between lightness and reflectance be-comes apparent, for it is known experimentally thatequal reflectance intervals do not correspond to equallightness intervals. For example, if the reflectance scale isdivided into 10 equal lightness intervals, the lowest willextend from 0 to approximately 1% on the reflectancescale, the next from 1 to 4%, and so on, with the lastinterval reaching from about 80 to 100%. The relation-ship is nearly logarithmic, in accordance with Fechner'slaw, but depends upon the background against which thesamples are viewed. For light backgrounds, lightnessdifferences accord approximately with differences inthe square roots of the reflectances.

The process of dividing the reflectance scale into equallightness intervals involves the average estimates oflarge numbers of observers.

Being thus equipped to establish a lightness scale,we inquire into how a chromaticness scale might beachieved. Again, the process is one of arraying all ofthe possible elements in accordance with some physi-cally measurable attribute and then marking offvisually uniform steps. It is thus possible to establisha system wherein each chromaticness is designated bya pair of numbers (chromaticity coordinates) and socan be located uniquely in a physical plane. The numberpair specifies the chromaticity of a color and arises froma pair of ratios between the amounts of three arbitrarilychosen primary colors which will combine to matchthe given color when viewed by a standard observer.Once the chromaticities are arrayed in an orderedmanner, the division into approximately equal chroma-ticness intervals is achieved by means of observerestimate.

It is a property of the human eye, however, thatexact uniformity of the chromaticness surface cannotbe achieved on a surface of zero curvature (plane). Forexample, if chromaticness is tentatively represented ona plane in polar coordinates, where hue varies withangle and saturation varies with radius, it is experi-mentally possible to divide the hue circle, or circum-ference, into an arbitrary number of visually equalsteps. One could also divide a diameter into a numberof saturation steps, each of the same perceptual size asthe hue steps. It would then be found, according toMacAdam,3 that the circumference does not containexactly r times as many steps as the diameter. Onemust thus accept the distortions in a Euclidean colorthree-space, hoping to render them small enough to beof little concern in most color problems.

Granting that the chromaticness plane may be made

3 D. L. MacAdam, J. Opt. Soc. Am. 33, 18 (1943).

approximately uniform, the remaining problem is theselection of a unit of lightness difference and a unit ofchromaticness difference such that these units will bevisually equal. It is perhaps not immediately obviousthat such a comparison is possible, but it has beenshown4 that very good agreement may be expectedamong observers in comparing the respective sizes of alightness difference and a chromaticness difference.Superficially, then, it would appear necessary merelyto invoke observer estimate again in order to adjustthe relative sizes of these units. But here we are besetwith further difficulty. The sensation difference evokedby a given pair of chromaticities of the same luminousreflectance does not remain fixed as their reflectance ischanged. It is clear, for example, that all chromaticitiesmust approach each other in chromaticness (that ofblack) as their reflectance approaches zero. Anotherdifficulty, which brings us to the matter at hand, is themanner in which the spatial proximity of a pair ofsamples affects the perception of their color difference.It is to be expected that a color difference will becomeless noticeable as the samples are moved apart. It isnot generally known, however, that it is the lightnessdifference which is largely suppressed, the chromatic-ness difference being relatively less affected. The resultis that the ratio of lightness-difference unit to chroma-ticness-difference unit is seriously affected by sampleseparation.

Thus, a given color space can have meaning forsamples of fixed size at one separation only. As theseparation is changed, so must the entire color solidshrink or stretch along the lightness axis.

Although proximity is only one among several viewingconditions which could be incorporated into a colorspace, it is of interest for several reasons. First, it is areminder that in any color-matching operation, thewidth of dividing line has an important effect on theaccuracy of matching. The difference between a narrowdividing line and none at all is clearly felt in the results.

A second reason for examining the proximity effecthere is that its selective action on the separate mecha-nisms for lightness and chromaticness discrimination isnot widely recognized. It is another detail which mustbe fitted into the structure of any satisfactory color-perception theory and may, indeed, be taken as evidencein support or denial of various existing theories.

JUDD COLOR SPACE

The proximity effect was described by Judd5 in 1939.It has been incorporated into his color-differenceformula which thereby specifies a color space whichshortens along the lightness axis as the spatial separa-tion of the samples is increased, and conversely. Judd'sstarting point was a chromaticity plane, his well-known

4 I. A. Balinkin, Am. J. Psychol. 52, 428 (1939).5 Deane B. Judd, Am. J. Psychol. 52, 418 (1939).

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JUDD COLOR DIFFERENCE FORMULA

uniform-chromaticity-scale triangle' within which equaldistances have approximately equal visual significance;that is, they correspond to approximately equal chro-maticness differences. At right angles to this plane wasadded a lightness axis, forming a color solid. The solidmay be pictured as a continuum of superposed chroma-ticness planes representing progressively lighter colorsin the positive direction of the lightness axis.

If AS represents a rectilinear length on the UCStriangle and AC is the corresponding chromaticnessdifference, the latter is assumed proportional both toAS and to the fourth root of the luminous reflectance,A, of the samples:

AC=k 2 A1 AS, (1)

where k2 is a normalizing factor.The lightness difference term AL is formed on the

basis that lightness correlates rather well with thesquare root of reflectance:

AL= kA(A). (2)

The proximity factor k1 adjusts the relative importancesof lightness and chromaticness differences as the sampleseparation is varied. In terms of our mental picture ofthe color solid, it is a weighting factor which stretchesthe lightness axis as a function of spatial proximity ofthe samples.

The geometry of the color space is assumed Euclidean,the net color difference, AE, being the magnitude of thevector sum of the lightness and chromaticnessdifferences:

(AE)2= (C) 2 + (AL)2

= [k2A iAS]2 +[kA (Ai)]2 . (3)

The ratio of k to k2 depends upon the relative im-portances of lightness differences and chromaticnessdifferences, whereas their sum establishes the size ofthe NBS unit of color difference, or the judd. The valuesuggested for k by Judd was 120 for samples in closecontact. To adjust for the proximity effect, k shoulddrop to 90 or 100 if a narrow dividing line separates thesamples, and to 30 or 40 if the samples are separated bya broad patterned area.

A value of 600 was suggested for k2 so that, in com-bination with the range 120 k l)30, the color differ-ence formula would predict a value of AE= 1 for colordifferences which were large enough to be perceptiblebut small enough to be negligible in most commercialtransactions.

The present research was undertaken in order tostudy the behavior of k and to arrive at an empiricalexpression for it as a function of the angular dividing-line width of a pair of samples under specified viewingconditions.

6 Deane B. Judd, J. Opt. Soc. Am. 25, 24 (1935).

EXPERIMENTAL PROCEDURE

Apparatus

An apparatus was constructed such that a pair ofcolored paper samples was presented to the observer inthe form of 1-cm squares side by side. Unrestrictedbinocular vision was used at a distance correspondingto an angular field of approximately 20. The illuminancewas approximately 20 ft-c.

The samples differed liminally in lightness but notmeasurably so in chromaticness. They appeared againsta dark background, of about 5% luminous reflectance,and were spaced a known distance apart by a stepwisevariable separator of the same reflectance as the back-ground. The samples were mounted on a small platewhich could be made to rotate in its own plane, whichwas vertical. The 1-cm squares were provided by apair of apertures in a mask which was pressed againstthe samples except when they were being spun. Changesin sample separation were effected by changing themasks.

At each presentation of the samples, the darker ofthe two might appear either to the right or to the left.The observer was asked to indicate the position of thedarker sample by pressing the proper one of a pair ofpush-button switches. His choice was immediately ad-judged "correct" or "incorrect" by the apparatus andrecorded as such by a pair of electromechanicalcounters. The mask was then released automaticallyand a mechanical impulse caused the sample plate tospin a few times and then come to rest randomly in oneof two stable horizontal positions. The mask was thenretracted, ending the one second cycle.

Figure 1 shows the apparatus with the cover andaperture mask removed. A schematic diagram is givenin Fig. 2.

Preparation of Samples

During the course of the experiments, severalmethods were evaluated for producing sample pairs

FIG. 1. The testing apparatus with cover and mask removedand with sample pair in position. Luminous reflectance differenceis not perceptible in photograph.

757July 1961

ALAN C. TRAUB AND ISAY BALINKIN

FIG. 2. Electromechanical details of apparatus. Timing platecycle begins when observer makes judgment and operates switch.Sliding contacts designated A, B, and C deliver programmedsequence of impulses to their respective solenoids. Mask solenoidsA release mask from contact with spinner plate. Feeler banksolenoid B (not shown) closes contacts which sense orientation ofrouting circuit plate, assess observer's judgment, and deliverimpulse to appropriate counter. Spinner plunger solenoid C thendelivers mechanical impulse to spinner plate shaft. At end of cycle,mask is again pulled in.

with liminal lightness differences and imperceptiblechromaticness differences. Pairs of graded Munsellcolor chips differing only in value were found to exhibitsupraliminal lightness differences under the viewingconditions used here. Another method involved thespraying of poster colors on smooth white paper. Thedarker half of each sample was achieved by adding atouch of black pigment before spraying, but this oftencaused a hue change which was difficult to cancel.

0

I-zw

a.

Ou~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~1

50°

40 o _ 0 -_ _______ g--40- . ~ - 0 - - - - - - ,-

30 /8

20 ,I

X - AVERAGE

0 - - - AA_U U 0.40

e IN DEGREES

Uniform rectangles of colored paper were then partlymasked and treated with a fine spray of black pigmentwithout much improvement over the mixture method.

Another method attempted was the selection ofsample pairs from a large number of commercial paintchips. Selections were made visually and examinedcalorimetrically on a Hunter multipurpose reflectome-ter. This method likewise failed to yield many satis-factory sample pairs.

The few successful sample pairs yielded by the abovemethods were used in the early tests. It soon becameapparent that neither the hue nor saturation of thesamples strongly affected the proximity function. It wasthereupon decided that achromatic sample pairs wouldbe quite satisfactory for the tests. Such pairs werereadily produced by exposing appropriately maskedsheets of photographic paper to controlled illumination.Uniform samples with any desired degree of reflectancedifference could thus be produced. The processed sheetswere calibrated on a Hunter color and color differencemeter and provided most of the data upon which theresults are based.

Data and Results

After each 100 judgments on a given sample pair ata given separation, the number of errors was recordedand the procedure was repeated for a new value of sepa-ration. This was continued until several values of sepa-ration had been used, in the range from 0 to about 3cm, after which a new sample pair was used. For eachpair, a curve was plotted showing percentage of errors

z

a.

Z

I-

03I-a.

0

a.

0.80 0.88

FIG. 3. Percent errors vs angular dividing linewidth, 0, fortypical sample pair. Each point represents 100 judgments. Crossesshow averages.

1.0 2.0 3.0

LIGHTNESS DIFFERENCE IN JUDDS

4.0

FIG. 4. Perceptibility vs lightness difference for zero dividinglinewidth. Based on 21 sample pairs, 3400 observations.

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JUDD COLOR DIFFERENCE FORMULA

versus angular separation 0 as subtended at the eyes.A typical curve is shown in Fig. 3. The number oferrors is seen to approach 50% as the separation in-creases and the judgment becomes completely random.

The scores were then converted into units of "per-ceptibility" on the assumption that, when undecided,the observer was equally likely to press either button,there being no "equal" category. While not strictlycorrect, this assumption is considered sufficiently accu-rate for our purposes.

The conversion is of the form "Perceptibility= 100%-2 (percent errors)," so that a score of zero errorswould indicate 100% perceptibility, whereas an errorscore of 50% would signify zero perceptibility.

The next step was to convert the vertical axis fromunits of perceptibility into units of lightness differencein judds. That is to say, we ask, "If a given reflectancedifference at a given separation has a given percepti-bility, what is the lightness difference in judds corre-sponding to this perceptibility?"

Our starting point is Judd's suggestion that theproximity factor be taken as 120 for zero separation.We then need merely measure A (A ') for a given samplepair, multiply by 120 to find the lightness difference injudds for zero separation, and then test that samplepair at zero separation to find the correspondingperceptibility.

This was done with 21 sample pairs on which 34 setsof 100 observations each were made, each set beingshown as a point in Fig. 4. The curve drawn throughthe points shows that, for zero separation, a lightness

0.40

e IN DEGREES

FIG. 5. Perceptibility curves for eight sample pairs. Units ofperceptibility have been converted to equivalent lightness differ-ence at zero separation from data in Fig. 4.

difference of 2 judd would be perceptible in only about3 observations in 100, 1 judd was visible about 50%of the time, and 4 or more judds were noticeable withcertainty.

The smoothed perceptibility curves for 8 samplepairs are shown in Fig. 5, the vertical axis having beenconverted from units of perceptibility to units of light-ness difference. These pairs were denoted by the lettersA through in the order of increasing reflectance dif-ference, pair A being the least different and hence theleast distinguishable.

The assumption is now made that perceptibility iscorrelated directly with lightness difference at sepa-rations other than zero, it being remembered thatlightness difference is regarded here as defined by acombination of a reflectance difference and a spatialseparation. Thus, a given reflectance difference at oneseparation may be perceptually equal to another at adifferent separation.

The meaning of Fig. 5 is the following. Each curvedenotes a sample pair with a known reflectance dif-ference. At zero separation, the lightness difference injudds for each pair is likewise known, assuming k1 equalto 120. If a horizontal line is drawn at any height acrossthe curves, each intersection will designate a pair ofvalues of reflectance difference and angular separation.Because all such pairs along the line resulted in equalperceptibilities, they are assumed to correspond to equallightness differences.

0.20 0.40e IN DEGREES

0.60 0.80

FIG. 6. Proximity factor vs 0 for eight curves ofFig. 5, assuming kl (0) = 120,

759July 1961

ALAN C. TRAUB AND ISAY BALINKIN

120

110*

0

C)

U-

X00L:

40-U)

0

z:

c, 0a:wja.

.----------------- -_ _ _ _ _ _ _ _ -

----v - ----- -- ---- --------- - - - - - - - - - --

10 r';,,* ~0 ---------- -- - - - - -_

0 0 0.20 0.40

e IN DEGREES

0.60 0.80 0.88

FIG. 8. Discrimination data for three sample pairsdiffering in chromaticity only.

where 0 is measured in degrees. The function is seen toapproach the value of 30 as an asymptote, which is notfar from the value suggested by Judd for the lower limit.

0.40

e IN DEGREES

FIG. 7. Average curve of Fig. 6 (dashed) compared withan analytical expression.

Figure 5 therefore contains the information necessaryfor finding an expression for the proximity factor as afunction of angular separation 0. By graphical solutionof the equation for lightness difference [Eq. (2)], eachcurve of Fig. 5 yields a curve of k (), as shown in Fig. 6.Ideally, one would expect these curves to coincide;their failure to do so is laid to experimental error andto observer fluctuations. The average of all the curvesis shown as a heavy line. It is shown again in Fig. 7,this time as a dashed line, where it is compared with aplot of the function

kI(0)=30+[18/(0.2+01)] (4)

Chromaticity Differences

A few data were taken on three sample pairs differingin chromaticity rather than in reflectance. The purposewas to determine whether, under the conditions of theseexperiments, the effect of proximity on the perceptionof chromaticity differences might be observable. Theresults are shown as error curves in Fig. 8 and indicatethat, compared with its effect on lightness differences,proximity did not play a large part in the recognition ofchromaticness differences. It is possible that other testconditions, such as a larger test field or a less contrastingbackground, might reveal such an effect.

ACKNOWLEDGMENT

It is a pleasure to express our grateful thanks toDr. Deane B. Judd for his suggestions regarding certainaspects of this presentation.

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